Questions on the use of algebraic techniques to prove geometric theorems.

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-1
votes
1answer
63 views

Find equations of two circles drawn through the origin which cut another circle orthogonally and touch a line

Find equations of two circles which are drawn through the origin to cut the circle $$x^2+y^2-x+3y-1=0$$ orthogonally and to touch the line $$x+2y+1=0$$. $$x^2+y^2-2ax-2by=0$$----(1) is the general ...
0
votes
1answer
40 views

Meaning of $\dashv\vdash$

I was looking at ProofWiki's articles 'Definition:Equidistance' and 'Definition:Between (Geometry)'and came across the symbol '$\dashv\vdash$.' What does it mean?
1
vote
1answer
34 views

Do we have $proj_u(a) + proj_u(b) = proj_u(a+b)$?

Let $a, b, u$ be vectors in $\mathbb{R}^3$. For two vectors $r, u$ in $\mathbb{R}^3$, let $proj_u(r)$ be the projection of $r$ on the line of $u$ in $\mathbb{R}^3$. Do we have $proj_u(a) + proj_u(b) = ...
3
votes
1answer
109 views

Intersection of a line through two points on a unit circle with real axis

Suppose we are given two points on unit circle which are represented as complex numbers $u$, $v$. We want to show that the intersection of the line through $u$ and $v$ and the real axis is ...
0
votes
0answers
52 views

Good book for Solid Analytical Geometry?

So my teacher uses this book, William H McCrea's Analytical Geometry of Three Dimensions, but it's awfully hard and dry. I need something with more exercises and better explanations, but that covers ...
0
votes
0answers
34 views

Is the unit circle uniquely defined by all it's sliced averages?

Suppose you have a unit circle $g(x,y) = x^2+y^2-1 = 0$ and for each $\theta \in (-\pi/2, \pi/2)$ you associate a map $f_\theta(c): R \to R^2$ defined by $$f_\theta(c) = \langle g^{-1}(0) \cap ...
3
votes
2answers
104 views

Equation for Tangent Line that passes through $(0,1)$ on the curve $y = \ln x$

I'm totally lost. I've been trying to figure this out. This is what I've figured out: $dy/dx = 1/x$ $y$-intercept $= 1$ So I try to do $y-y_1 = m(x-x_1)+b,$ which I get as $y-1 = 1/x(x-0)+1,$ ...
0
votes
2answers
62 views

Finding the length from an interior point of a triangle to a vertex given distances to the other two

So let's assume that there is a triangle ABC and there is a point P inside of ABC. You are given the distances of AP and BP and you are trying to solve for CP. I faintly remember reading something ...
4
votes
4answers
99 views

Find the equation of a circle, given a point on it and a point where it is tangent to a given line

The given question is: Find the equation of the circle that passes through point $(-3,-4)$ and touches the line $x-y+7=0$ at the point $(-5,2).$ What I did was: Took the given points $(-5,2)$ and ...
-3
votes
1answer
18 views

What is the domain of the given function with the greatest integer?

The domain of the function $$f(x)=\sqrt{\frac{4-x^2}{[x]+2}}$$ where $[x]$ represents the greatest integer function, is (a) $(-\infty,-1)\cup[-1,2]$ (b) ...
0
votes
0answers
34 views

Arc measures in a circle

Suppose we have a quadrilateral inscribed in a circle prove that angles inside the same arc are equal
6
votes
1answer
103 views

A parabola lemma

I am looking for a previous reference and/or a geometric proof of the following lemma: Let $P$ be the parabola $y=x^2$. Let $a$, $b$, $c$, $d$ be four points on $P$ sorted from left to right, and let ...
2
votes
1answer
32 views

Line with predefined length tangent to circle

I have one math problem which I'm trying to solve. I know it could be done but I'm a little bit "rusty" with my algebra. I'm kindly asking for help. Problem and procedure of my solution are shown in ...
0
votes
1answer
19 views

Find the vectorial equation of the line through $P$ and orthogonal to two planes

I have to find the vectorial equation of the line through $P$ and orthogonal to $r:(x,y,z)=(1,−1,−1)+\lambda(1,−1,0)$ and $s:(x,y,z)=(\frac{3}{2},-\frac{1}{2},0)+\alpha(\frac{1}{2},\frac{1}{2},1)$. ...
0
votes
2answers
34 views

Compute position of next point on a line

I'm writing a program in which it is possible to draw a horizontal, vertical or an oblique line. So the line can be described as follows : $f(x) = y = mx + q$ But my problem is that given the first ...
2
votes
4answers
115 views

Area of triangle bounded by line and degenerate “crossed lines” conic

The question is Show that the two lines given by $$(A^2 - 3B^2)x^2 + 8ABxy +(B^2 - 3A^2)y^2=0$$ and the line given by $$Ax+By+C=0$$ determine an equilateral triangle of area ...
1
vote
1answer
28 views

How to find the curve and axis?

We consider the following one-sheeted hyperboloid: $y^2-4x^2+4z^2=4$ This is also a surface (or solid) of revolution. So it must be generated by rotating a curve about an axis. What curve and axis ...
1
vote
1answer
42 views

How can I multiply by time?

I'm reading this article about collision detection. In it, he says: However, t appears to be referring to time - The time of ...
0
votes
1answer
22 views

Derive the 2-D analogue of the Laplace Dispersal Kernel using RDE

I found an interesting problem. I'm looking at the Laplace Dispersal Kernel for 1 dimensional dispersal behavior. And I wonder what happens in two dimensional world? I managed to find the limiting ...
0
votes
0answers
51 views

Analytical solution to nonlinear ode

I solved this equation that I attached numerically in matlab by the Newton Raphson method. Now I want to solve it analytically in matlab or even in Maple if it is possible. Would you please help me ...
0
votes
2answers
55 views

Linear algebra - find all possible positions of the third corner?

An equilateral triangle lies in the plane $x + y - z = 1$ and corners in points $(1, 1, 1)$ and $(2, 1, 2)$. Determine all possible positions of the third corner?
6
votes
4answers
111 views

finding the max of $f(x)=\sqrt{(x^2-4)^2+(x-5)^2}-\sqrt{(x^2-2)^2+(x-1)^2}$

I need to find the max of $$f(x)=\sqrt{(x^2-4)^2+(x-5)^2}-\sqrt{(x^2-2)^2+(x-1)^2}$$ When $x$ is a real number. What i did is to simplify: $$f(x)=\sqrt{x^4-7x^2-10x+41}-\sqrt{x^4-3x^2-2x+5}$$. Then ...
2
votes
2answers
102 views

Area of ellipse not in xy-plane

I've got a problem in which I'm trying to find the area of an ellipse which is given by the intersection of an elliptic cylinder with a plane. Nothing here is parallel to the coordinate axes, which is ...
0
votes
1answer
66 views

Finding equation of ellipse with given point and distance between directrices

I need to find the equation of an ellipse. The given were just a point where it passes, and distance between directrices. I know that the distance between directrices is given by $2a/e$. I don't ...
0
votes
1answer
32 views

Find the equation of a hyberboloid with given base, narrowest section, and the distance between them

I have one question left in an assignment and I havn't been able to solve it. I know the equaton for a hyperboloid and I know that $a$ and $b$ will be equal to each other. I don't know how to solve ...
1
vote
3answers
32 views

Finding circle of a sphere through two points

We have two points $P_1, P_2$ on a sphere $S$ of radius $R$. Suppose for $r \ll R$, the distance between $P_1$ and $P_2$ is less than $2r$. Then, $P_1$ and $P_2$ both lie on exactly two radius-$r$ ...
1
vote
1answer
74 views

Intersecting two parabolas and computing the angle between the tangents in a point of intersection

I was solving some problems on parabola. I saw a question and solved it, but my solution was way too big. The question was: If $$\left(\frac{a}{b}\right)^{1/3}+\left(\frac{b}{a}\right)^{1/3} = ...
1
vote
0answers
63 views

Proof 5 points determine a conic without projective geometry

So I'm trying to prove that any five points, of which no 3 are colinear, there is a single conic that passes through al of them. I don't want to use projective geometry but rather, only analytic ...
1
vote
4answers
136 views

How to tell whether a point is to the right or left side of a line

I have a line equation in the form ax+by+c=0 and a point p(x,y).How can I determine on which side of the line the point is located?
0
votes
1answer
24 views

Graphing and finding sine wave info with $\sin(x/4)$

I'd doing a chapter on graphing sine waves, and finding the amplitude, period, and so on. I know something like $y = 2 \sin(3x+ \pi) + 1$ can be turned into $y = 2 \sin[3(x + \pi/3)] + 1$ following ...
0
votes
0answers
37 views

Prove $(\vec{r}\cdot\nabla)\vec{u}+\vec{r}\times (\nabla \times \vec{u})=\vec{r}(\nabla\cdot\vec{u})+(\vec{r}\times \nabla )\times\vec{u}$

Show that $$(\vec{r}\cdot\nabla)\vec{u}+\vec{r}\times (\nabla \times \vec{u})=\vec{r}(\nabla\cdot\vec{u})+(\vec{r}\times \nabla )\times\vec{u}$$ I have been trying to show this for the past few ...
0
votes
2answers
47 views

How to connect a line between 4 randomly placed points on a plane such that the line does not cross itself

You get 4 coordinates of points on a plain. You need to connect them all with a line. The line must not cross itself. What's your strategy?
0
votes
1answer
32 views

Equation of a parabola: Translations and rotation

I've tried to solve this problem: Find an equation of the parabola with vertex at point $(1,1)$ whose directrix is the line $x-2y=6$. It has to be solved using translation and rotation (coordinate ...
1
vote
1answer
51 views

Find the distance of a point from a plane generated by two given vectors

I need to calculate the distance of the point $P = (0, 5, -4)$ from the plane which pass from the point $P1=0, 1, -2)$ and generated by the two vectors: $$ v1 = (1, 2, 3), v2 = (-1, \sqrt{2}, 1) $$ ...
0
votes
0answers
60 views

What is the best book to learn coordinate geometry

The level should be above high school, and it must be free online if at all possible. Also, I have an additional question to ask: How many months, roughly, would it take to finish a mathematics book ...
0
votes
2answers
73 views

What is the minimum information required to define an equation for ellipse?

What is the minimum information ie. amount of points in 2-dimensional plane in order to define the equation for an ellipse? I know that unique ellipse cannot be defined when only one of the foci is ...
0
votes
0answers
27 views

What is the best book for coordinate geometry?

Requirements: A) not too thick , as I am reading this only to solve calculus problem . B) free on web is the best (optional) C) I don't mind if the book involves both coordinate geometry and ...
1
vote
2answers
55 views

Can ellipse equation be transformed through one of its foci?

Can we transform ellipse equation to represent an ellipse transformed by tilting it through its focus such that its center point moves in circular manner and one of its focus stays at constant ...
1
vote
2answers
78 views

Is it possible to find equation for ellipse when focus, eccentricity and two points are known?

Is it possible to find equation for an ellipse when we know two points and one focus in 2d cartesian coordinate system? We can also make these assumptions about these two given points depending on ...
0
votes
1answer
71 views

How can I find the common axis of 2 cones in space that have the same base radius but different heights?

How do I find the 3D vector describing the axis of 2 overlapping cones, like this: If I have only the following information: Coordinates of the common tip Coordinates of a point on the yellow ...
2
votes
0answers
17 views

Find a vector equation and parametric equations for the line segment that joins $P$ to $Q$. [duplicate]

Find a vector equation and parametric equations for the line segment that joins $P$ to $Q$. Here $P(1,-1,7)$ and $Q(7,5,1)$. I have tried to find $r(t)$ by using the formula $r(t)=p+t(p-q)$ but ...
0
votes
2answers
20 views

Why $(h,k)$ in equation $y= a(x-h)^2 +k$ is the vertex of a parabola?

As in the title , I know how to convert normal explicit equation to a vertex form equation by completing the square . But what is the reasoning behind why $(h,k)$ must be the vertex , but not other ...
0
votes
2answers
48 views

What does 'forms a right-handed set' mean?

In a question I am reading, the following question appears. What if $\vec{A},\vec{B}$, and $\vec{C}$ are mutually perpendicular and form a right-handed set? What exactly does "form a ...
1
vote
2answers
35 views

Hyperbola question

the graph $ y^2=16x $ is a hyperbola; it can be rewritten as $ y= \pm 4\sqrt{x}$ when I draw it down however It is clearly not a function..question is whether it has to be one in order to perform ...
0
votes
1answer
17 views

What is the vector equation of the line through the head of $v_0$ and parallel to $v_p$?

$v_0$ and $v_p$ are vectors. Let $v_0, v_1$ and $v$ be vectors, all emanating from $(0, 0, 0)$. Suppose the line $l$ is passing through their heads. Let $v_p$ be on the line $l$ such that $v_1 = v_0 ...
2
votes
0answers
33 views

About parametric equation of a line in $3$-space

$a.$ Given coordinates $(x, y, z )$ with origin $(0,0,0)$, parameterize the line through the points $(4,5,6)$ and $(1,2,3).$ $b.$ Take components of your answer to Part $(a)$ to give three ...
0
votes
0answers
30 views

Endless repeating tiles

Regarding to this question ( "Hall of mirrors" OR "wraped plane" - Problem ) there's still an open point: how can i determine if the sum of all vectors pointing from p1 to p2 is ...
1
vote
2answers
21 views

How to find the equation of $f(x^3)$ given the tangent in $f(x)$? [closed]

The exercise says: The tangent of $f(x)$ in $x=1$ is $y=2x-1$. Find the tangent of $y=f(x^3)$ in $x=1$.
2
votes
1answer
19 views

finding a soild from five planes

Given five planes: $\pi_1=2x+5y+z-2=0$ $\pi_2=x+y-z-1=0$ $\pi_3=x+4y+2z-4=0$ $\pi_4=3x-y+4z-3=0$ $\pi_5=-6x+2y-8z+k=0$. How can i find the solid shape that is formed by those planes? I tried to ...
0
votes
2answers
48 views

About the sum of sines of two angles

Suppose that $0\le \alpha\le \pi/2$ and $0\le \beta\le \pi/2$ such that $\alpha+\beta\ge \pi/2$. Can we prove that $\sin(\alpha)+\sin(\beta)\ge 1$?